how to keep up with mathematics

How to Keep Upwith MathematicsThird Kenneth C. Schraut

Memorial Lecture

University of Dayton

November 2, 2002

Paul J. Campbell ’67Beloit College

Beloit, WI 53511

[email protected]

See . . . Seek . . . Speak . . . Mathematics

Outline

•SEE . . .

•SEEK . . .

•SPEAK . . .

mathematics

1


SEE mathematics . . . :

serious applications

•Error-correcting codes

(book numbers, UPCs, driver’s

licenses, banknotes)

•Benford’s law

•CDs and MP3s

•Web searching

2


ISBN NumbersExample:

0 – 7167 – 4782–︷︸︸︷country

︷︸︸︷publisher

︷︸︸︷book

︷︸︸︷check digit

For a correct ISBN a1a2 · · · a10,

10a1+9a2+8a3+7a4+6a5+5a6+4a7+3a8

+2a9 + 1a10 ≡ 0 mod 11, or

a10 ≡ 1a1 + 2a2 + 3a3 + 4a4 + 5a5

+ 6a6 + 7a7 + 8a8 + 9a9 mod 11

= remainder after division by 11= 1 · 0 + 2 · 7 + 3 · 1 + 4 · 6 + 5 · 7

+ 6 · 4 + 7 · 7 + 8 · 8 + 9 · 2

= 0 + 14 + 3 + . . . + 18

= 231 = 0 + 21 · 11 ≡ 0 mod 11

When the remainder is 10 . . .

The ISBN detects all transposition errors, all singleerrors—90% of all errors.

3


Benford’s LawFrequency of leading digit d = 1, 2, . . . , 9:you expect 1/9, but for many data sets get:

log10(1 + 1/d)

log10

(1 + 1

1

)≈ .3, . . . , log10

(1 + 1

9

)≈ .05

!!!

1881: Simon Newcomb: log tables1938: Frank Benford: areas of rivers,

atomic weights, electric bills, . . .—analogous laws for second, third, etc.

significant digits

Why?

1995: Ted Hill (Georgia Tech) proved—unique base-invariant distribution—unique scale-invariant distribution—random samples from random

“neutral” data sets

—applications: fraud detection;computer design

4


CDs and MP3 PlayersPulse Code Modulation:

—samples sound (voltage) 44,100 /sec—gives 16-bit digital result—for one-minute of stereo: 8.1 MB—CD: 650 MB/8.1 MB/min ≈ 80 min—error-correction overhead

(Reed-Solomon code) ⇒ 74 minCompression:

—PK-ZIP, Stuffit: reduce only 10%—special algorithms: reduce 60%

Lossless =⇒ “lossy” compression1992: MP3 (= MPEG3): compression

ratio from 8 to 12—divides frequency range into 32 bands—modified discrete cosine transform

gives 18 constituents/band—where the loss comes in: “masking,”

removing redundancy in each band—Huffman coding compression

(shorter codes for frequent values)

5


Web SearchingWeb site: important if others link to it

x1, x2, . . . , xn the importances

Idea: importance of site ∝ sum ofimportances of sites that link to it

e.g.,x1 = k(x14 + x97 + x541)

Construct n × n matrix A:

ai,j =

{1, if site j links to site i;

0, else.

xi = kn∑

j=1

ai,jxj (i = 1, . . . , n)

as matrices: X = kAX , AX = 1k

X

—an eigenvalue/eigenvector problem!

—Google uses a variant: n = 2.7 billion!!!

eigenvalue ranking: Kendall, Wei 1950s

another application: “top ten” teams

6


SEE mathematics . . . :

fun examples

•Matching problems

(gift, card game problems)

•Dynamic programming

(Farmer Klaus, football)

•The Wave

•Markov chain models

(baseball, Monopoly, . . .

7


Matching ProblemsThe Secret Santa, or Guy Gift, Problem

(aka The Hat-Check Problem)P (no match) ≈

(1 − 1

n

)n → e−1 ≈ .37

The Guy Gift Problem with CouplesP (no match) → e−2 ≈ .14

The Bridge Couples ProblemP (no match) → e−1/2 ≈ .61

p(k) →e−λλk

k!Poisson, λ = 1/2

The Card Game of WarP (no “battles”) → e−3/2 ≈ .22

p(k) →?? Poisson, λ = 3/2

P (“annihilation”) = e−3/2/226 ≈ 3 × 10−9

A Different Matching ProblemAlgorithm for Matching

Medical Residents to Hospitals(aka The Marriage Problem)

8


Farmer Klaus and the MouseGoal: Get all the grain in(6 sacks each of 4 kinds)before 6 mice come outof their holes.

6-sided die:each kind of grain, a mouse,and a generic sack

What is P (children win)?

Method: Monte Carlo simulation?Better here: dynamic programming,

using symmetry to reduce size

• Expand out along the game tree

• Solve each simple case only once, rememberresult

• Apply that result as the simple case arises overand over again

• Propagate back the results from simple cases

9


Coin Flipping

You’re ahead 1–0.What is your chance of winning?

3–0✐

3–1✐

3–2✐ 2–3✐2–2✐

2–1✐2–0✐

2–1✐

2–2✐ 1–3✐1–2✐

1–1✐1–0✐

1/2 1/2

1/2 1/2 1/2 1/2

1/2 1/2 1/2 1/2

1/2 1/2

10


1❦✐

1❦✐

1❦✐ 0❦✐2–2✐

2–1✐2–0✐

2–1✐

2–2✐ 0❦✐1–2✐

1–1✐1–0✐

1/2 1/2

1/2 1/2 1/2 1/2

1/2 1/2 1/2 1/2

1/2 1/2

11


1❦✐

1❦✐

1❦✐ 0❦✐1/2❦✐

3/4❦✐7/8❦✐

3/4❦✐

1/2❦✐ 0❦✐1/4❦✐

1/2❦✐

11/16�❧1/2 1/2

1/2 1/2 1/2 1/2

1/2 1/2 1/2 1/2

1/2 1/2

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Simplified versions: Pascal’s“problem of the points”

With best play, P (children win) =

25693777239213422058225192034489

50209695374400000000000000000000≈ .512

Other Examples ofDynamic Programming

Obstgarten, . . .Shut the Box

variants on Blackjackmancala gamesfootball—point-after-touchdown strategy—fourth-down strategy (August 2002)

13


The Wave

1981: first(?) appearance, atOakland Athletics game

20 seats/sec with ave. width of 15 seats

majority right-handed ⇒ clockwise

can apply mathematical models for—spread of forest fires—propagation of electrical impulses

in heart tissue

14


September, 2002: Model of the wave

activation probability—decreases exponentially with distance—decreaseslinearly with cos(direction)

(so people in front, on left have moreinfluence)

affiliations of authors:—Dept. of Biological Physics—Institute for Economics and Traffic

15


Baseball and Markov ChainsHow much difference does

batting order make?

The state space of baseball:—18 half innings—in each half inning, 24 states:

{0, 1, or 2 outs} × {8 base-runner situations}—for each batter, 12 possible ball counts

Associate probabilities with transitions;get Markov chain model

Different results for different lineups

9! = 362, 880 lineups of 9 batters

Findings:—Place best batter 2nd, 3rd, or 4th.—Either the second-best or third-best

should go just before or just afterthe best batter

—Best lineup vs. worst: 4 to 5.5 gamesper season.

16


Markov Chain Theory—involves matrix algebra,

eigenvectors—determines average duration of game—determines average time spent in a

particular state

Other Examples of Markov Chainsgames

—Monopoly

—Chutes and Ladders

—Bingo

—Cootie

number of games in a World Series

. . . plus multitudes of seriousapplications(e.g., to social mobility)

17


SEEK mathematics . . . :•serious mathematics–mathematical

organizations–general press–mathematics indexes–mathematics

repositories–2002 examples:∗Catalan conjecture∗primality

18


How to Stay in Touch—passively: mathematics comes to you—math’l organizations and their publ’ns

Math’l Ass’n of America, Amer. Math. Soc.,Ntl Council of Tchrs of Math,Soc. Ind’l and Appl. Math. (SIAM)

—general press and general science pressyour local paper, New York Times, Discover,Science, Scientific American,Science News, Nature

—more actively, for serendipity:—“columns” at Web sites of MAA, etc.

—when you’re hungry for more:—books in the library—mathematics indexes

MathSciNet, MathDI Database, Zentralblatt—mathematics repositories

Mathematics ArXivJSTOR

19


Heuristic for Finding Articles“Ask, and you shall be answered;

seek, and you shall find;knock, and it shall be opened unto you”

–Sermon on the Mount

1. Input author and topic into—general search engine (Google, NYT, . . . )—your library’s electronic journals index—mathematics indexes—mathematics repositories

2. Follow leads to—article summary or full text—reference to holdings in local libraries—author’s home page—journal’s home page

3. If no success:—email/write/fax/call author, locating from

home pages, Combined Membership List,World Directory of Mathematicians

—consult a librarian

20


Mathematical Advances in 2002:Polynomial-Time Primality TestWhy important?

contemporary cryptography requires primeswith hundreds of digits

Testing whether an integer n is prime:—brute force: try every prime ≤ √

n; takes timeexponential in log n (the number of digits)—i.e., time proportional to nnotation: O(n), “big-oh of n”

—ideal: polynomial-time algorithm “in P”in the number of digits: O

((log n)k

)—conditional algorithms: “If RH is true . . . ”—used in practice: probabilistic algorithms;

fast, but could be wrong

August, 2002: Agrawal with two undergrads(Indian Institute of Technology)show in brief paper that “PRIMES is in P”:almost O

((log n)12

), cond’ly O

((log n)6

)Speed up their algorithm, find faster ones?

21


Mathematical Advances in 2002:Catalan’s Conjecture ProvedWhy important?

it’s not, except to alleviate curiosity

Eugene Charles Catalan (1814–1894) conjecturedin 1844 that:

the only solution to xm − yn = 1in integers x, y, m, n ≥ 2 is 32 − 23 = 1

Main approach to both Catalan’s conjecture andFermat’s Last Theorem abstract algebra

—Ernst Kummer (1810–1893) used algebra toprove FLT for various classes of exponents

—20th century: new paradigm; both problemslead to elliptic curves (algebraic geometry)

—1995: elliptic curves are the key to FLT . . .— . . . but are dead end for Catalan’s conjecture

April, 2002: Preda Mihailescu (University ofPaderborn, Germany) proves conjecture bygoing back to Kummer’s algebraic theory

22


SEEK mathematics . . . :

• fun mathematics

–Martin Gardner:

books, indexes

–other books on

recreational math

–current sources

23


Mathematics for Fun—books by Martin Gardner

23


Sample Research Project:Optimizing Bingo

•SEEing Bingomathematically

–P (card wins)

–P (card wins on draw k)

–average duration:E [draws until a winwhen n cards in play]

–P (more than one winnerwhen n cards in play)

24


•SEEKing Bingo

–at the public library

–on the Web

∗history of Bingo

∗practical questions

∗mathematical research

questions

–at the bingo hall

– in the math literature

25


Mathematical Questions

•How large can a set of cards

be that always has a single

winner?

•How can you construct such

a set?

•With actual cards used, what

is the probability of a single

winner with n cards in play?

26


Some Approaches

•Heuristic: “Solve a simpler

problem”: 3 × 3 bingo

•Heuristic: “Translate the prob-

lem into various branches of

mathematics”

–geom.: intersecting lines?

–algebra: equations?

–comb.: Latin squares?

27


SPEAK mathematics . . . :

•don’t keep what you see to

yourself!

•don’t talk technical

•get a good coffee-table math

book

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See . . . Seek . . . Speak . . . Profit from . . . Mathematics

How to Make Big Money from Math

Theorem (Allis 1988, master’s thesis).The first player in Connect-4 can always win.

1. Learn the strategy(and teach it to your computer).

2. Learn German.3. Travel to Augsburg, Germany.4. Tune in the Munich TV station that lets you

play the game for thousands of dollars.5. Dial in fast and often (use your computer)

(costs $1 per call).6. Be lucky and get through.7. You get to play first, so beat the host.8. Stay on the line to tell them where to send

the money.

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Accompanying Handout

ftp://cs.beloit.edu/math-cs/Faculty/

Paul%20Campbell/Public/Schraut/

30

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